Inscribing Circles

This graphic should be self-explanatory, as to why the blue circle has less area than the green circle. Note that the big blue circle has the same area as four of the small blue circles which have half the diameter.

There is a quick way of knowing the fact by visualizing the picture. Keeping another identical green quarter at bottom left corner of figure (i) completely completes the diagonal length while small area is left out in figure (ii) which implies that area of green circle must be greater

---

Here is the lengthier way, where we figure out the lengths.

Here is the a quarter of each square.

Let the length of each quarter be x .

In fig (i),

AB = x $\sqrt{2}$

Diameter (pink) = x

Also,

Since there is space for another green circle quarter in the bottom left

Diameter (pink) + 2 Radius (green) = AB

x + 2 Radius (green) = x $\sqrt{2}$

Radius (green) = $\frac{x\sqrt{2}-x}{2}$ -----------------------------------(i)

In fig (ii)

AB = x $\sqrt{2}$

Radius (pink) = x

Since the circle doesn't touch the vertex of square

Radius (pink) + Diameter (blue) < AB

x + 2 Radius (blue) < x $\sqrt{2}$

Radius (blue) < $\frac{x\sqrt{2}-x}{2}$ --------------------------------------(ii)

From (i) and (ii),

Since, Radius (blue) < Radius (green)

So, Area of green circle > Area of blue circle